Finite Group Databasebeta

← #46846602

Subgroups — #46846602

Order 34688.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774371C211yesmaximal, characteristic, complemented
2#31774218C212yesmaximal, complemented
3#31774194C212yesmaximal, complemented
4#31774089C212yesmaximal, complemented
5#31774030C212yesmaximal, complemented
6#31774007C212yesmaximal, complemented
7C4×#16864639C212yesmaximal, complemented
8#31773958C211yesmaximal, characteristic, complemented
9#540×C271C211yesmaximal, characteristic, complemented
10C22×D1084C2211yescharacteristic
11C2×#16864639C2212yes—
12C2×#16864639C2212yes—
13C4×D1084C2212yes—
14C4×D1084C2212yes—
15#22576122—22no—
16#22576119C2211yescharacteristic
17C22⋊C4×D271—22no—
18#22576093C2212yescomplemented
19#22576091C2212yescomplemented
20C4⋊(C271⋊C8)C2212yescomplemented
21#22576081C2212yescomplemented
22#22576081C2212yescomplemented
23#22576080C2212yes—
24#22576080C2212yes—
25C22⋊(C271⋊C8)C2212yes—
26#22576071C2211yescharacteristic, complemented
27C4×C271⋊C8C2211yescharacteristic, complemented
28#127×C271C2211yescharacteristic, complemented
29#125×C271C2211yescharacteristic, complemented
30#123×C271—22no—
31#121×C271—22no—
32#121×C271C2212yes—
33C4×D4×C271C2212yescomplemented
34#117×C271C2211yescharacteristic
35C23×D271—21no—
36C2×D1084—24no—
37C2×D1084—21no—
38C2×D1084C2312yes—
39C2×C4×D271—22no—
40C2×C4×D271—22no—
41#16864639—24no—
42#16864639—24no—
43C2×C271⋊C8C2312yes—
44C2×C271⋊C8C2312yes—
45C4⋊Dic271C2312yes—
46C22⋊Dic271—21no—
47(C2×D271)⋊C4—22no—
48(C2×D271)⋊C4—22no—
49C4×Dic271—21no—
50C2×D4×C271—22no—
51C2×D4×C271—22no—
52C2×D4×C271C2312yes—
53C22×C4×C271—22no—
54C22×C4×C271C2311yescharacteristic
55C4⋊C4×C271—22no—
56C4⋊C4×C271D412yes—
57C4⋊C4×C271C2312yes—
58C22⋊C4×C271—22no—
59C22⋊C4×C271—22no—
60C22⋊C4×C271D412yes—
61C42×C271C2311yescharacteristic
62C42×C271C2311yescharacteristic
63C22×D271—22no—
64C22×D271—24no—
65C22×D271—82no—
66D1084—24no—
67D1084—22no—
68D1084#4114yes—
69C2×Dic271—22no—
70C4×D271—82no—
71C4×D271—82no—
72C271⋊C8—22no—
73C271⋊C8—22no—
74C23×C271—22no—
75C23×C271C2×D411yescharacteristic
76D4×C271—22no—
77D4×C271—22no—
78D4×C271—82no—
79D4×C271—82no—
80C2×C4×C271—22no—
81C2×C4×C271—22no—
82C2×C4×C271—82no—
83C2×C4×C271C2×D412yes—
84C2×C4×C271C2×D412yes—
85C2×C4×C271C2×D411yescharacteristic
86C2×C4×C271C2411yescharacteristic
87C22×C271—21no—
88C22×C271—22no—
89C22×C271—42no—
90C22×C271—42no—
91C22×C271—42no—
92C22×C271C22×D411yescharacteristic
93C2×D271—24no—
94C2×D271—42no—
95C2×D271—42no—
96C2×D271—84no—
97C4×C271—21no—
98C4×C271—22no—
99C4×C271—22no—
100C4×C271—42no—
101C4×C271C2×#4112yes—
102Dic271—42no—
103C2×C271—41no—
104C2×C271—82no—
105C2×C271#52511yescharacteristic
106C2×C271#57612yes—
107D271—44no—
108C271#312211yescharacteristic, complemented, Sylow
109#3122—2711nomaximal, Sylow
110#540D27111yescharacteristic, complemented
111#518—2711no—
112#482—2712no—
113#478—2712no—
114#459—2712no—
115#447—2712no—
116#442—2712no—
117C4×D8—2712no—
118#433—2711no—
119C22×D4—2711no—
120C2×D8—2712no—
121C2×D8—2712no—
122#127C2×D27111yescharacteristic, complemented
123#125C2×D27111yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D27112yes—
127C4×D4—2712no—
128C4×D4—2712no—
129C4×D4—5422no—
130C4×D4C2×D27112yescomplemented
131#117—2711no—
132#117C2×D27111yescharacteristic
133C2×C22⋊C4—5422no—
134#107—2712no—
135#106—2712no—
136C4⋊C8—2712no—
137#102—2712no—
138#102—2712no—
139#102—2712no—
140#102—2712no—
141C22⋊C8—2712no—
142#97—2711no—
143C4×C8—2711no—
144C24—5421no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—2712no—
148C2×D4—5424no—
149C2×D4—5421no—
150C2×D4C22×D27112yes—
151C22×C4—22no—
152C22×C4—5422no—
153C22×C4—5422no—
154C22×C4C22×D27111yescharacteristic
155D8—5424no—
156D8—5424no—
157C2×C8—2712no—
158C2×C8—2712no—
159C4⋊C4—22no—
160C4⋊C4—2712no—
161C4⋊C4C22⋊D27112yes—
162C4⋊C4C22×D27112yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—5421no—
166C22⋊C4—5422no—
167C22⋊C4—5422no—
168C22⋊C4C22⋊D27112yes—
169C42—5421no—
170C42C22×D27111yescharacteristic
171C42C22×D27111yescharacteristic
172C23—22no—
173C23—5424no—
174C23—5422no—
175C23—21682no—
176C23C2×C22⋊D27111yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—2714no—
182D4—5424no—
183D4—5422no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—5422no—
188C2×C4—21682no—
189C2×C4—21682no—
190C2×C4C2×C22⋊D27111yescharacteristic
191C2×C4C2×C22⋊D27112yes—
192C2×C4C2×C22⋊D27112yes—
193C2×C4C23×D27111yescharacteristic
194C8—5422no—
195C8—5422no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—5424no—
202C22—10842no—
203C22—10842no—
204C22—21684no—
205C22C22×C22⋊D27111yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—10842no—
211C4#41×D27112yes—
212C2—41no—
213C2—82no—
214C2—10844no—
215C2#3177440411yescharacteristic
216C2#3177470012yes—
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#540×C271C2
socleC22×C271C22×D4
hypercenter#540D271
hypocenterC271#3122
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #31774371C2semidirect
2. #31774218C2semidirect
3. #31774194C2semidirect
4. #31774089C2semidirect
5. #31774030C2semidirect
6. #31774007C2semidirect
7. C4×#16864639C2semidirect
8. #31773958C2semidirect
9. #540×C271C2semidirect
18. #22576093C22semidirect
19. #22576091C22semidirect
20. C4⋊(C271⋊C8)C22semidirect
21. #22576081C22semidirect
22. #22576081C22semidirect
26. #22576071C22semidirect
27. C4×C271⋊C8C22semidirect
28. #127×C271C22semidirect
29. #125×C271C22semidirect
33. C4×D4×C271C22semidirect
108. C271#3122semidirect
110. #540D271semidirect
122. #127C2×D271semidirect
123. #125C2×D271semidirect
130. C4×D4C2×D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317743712
#317742182
#317741942
#317740892
#317740302
#317740072
C4×#168646392
#317739582
#540×C2712
#3122271